Weighted Norm Inequalities for the Local Sharp Maximal Function

The Journal of Fourier Analysis and ApplicationsVolume 10, Issue 5, 2004Weighted Norm Inequalitiesfor the Local Sharp MaximalFunctionA.K. LernerCommunicated by Hans G. FeichtingerABSTRACT. A weighted norm inequality for the local sharp maximal function M λ # f is proved.Our main result along with the extrapolation theorem by D. Cruz-Uribe and C. Pérez is appliedto obtaining several new weighted norm inequalities for maximal functions and singular integrals.Several open problems are given.1. IntroductionThis note is inspired by the following extrapolation theorem of D. Cruz-Uribe andC. Pérez [4].Theorem A.Let S and T be operators (not necessarily linear) and let f be a function in a suitabletest class for both S and T . Suppose that there exists a positive constant c 0 such that forall weights ω∫∫R |Tf|ωdx ≤ c n 0 |Sf |Mωdx .nRThen for all p, 1

Weighted Norm Inequalities for the Local Sharp Maximal Function 467We now state the main result of the article.Theorem 1.For any measurable function f with f ∗ (+∞) = 0 and all weights ω,∫∫R |f(x)|ω(x)dx ≤ c n nR M# n λ nf(x)Mω(x)dx , (1.6)where constants c n ,λ n depend only on dimension n.Observe that it is easy to show that the condition f ∗ (+∞) = 0 is equivalent to thatthe distribution function µ f (α) =|{x ∈ R n :|f(x)| >α}| is finite for all α>0.Our proof of Theorem 1 is close in spirit to the proof of the following duality inequality[13, p. 146]:∫∣ ∫∣∣∣ ∣R f(x)g(x)dx ≤ cn R f # (x)Mg(x) dx , (1.7)nwhenever f ∈ L ∞ and g ∈ H 1 , where Mg is the grand maximal function. The proofof (1.7) is based on the atomic decomposition for the Hardy space H 1 , and on controllingoscillations of f by f # . Our proof consists of similar two steps as well. But since ωis a weight, we use a sort of atomic-like decomposition based on the standard Calderón–Zygmund decomposition. On the second step we use a sharper argument to control theoscillations of f . These yield the following improvement of (1.7):∫∣ ∫∣∣∣ ∣R f(x)g(x)dx ≤ cn R M# n λ nf(x)Mg(x) dx , (1.8)whenever f ∈ L ∞ and g ∈ H 1 . Note that, in view of (1.3), f # may be essentially largerthan Mλ # f . Observe also that it is crucial here to decompose into atoms whose size iscontrolled by their L ∞ norms. In the one-dimensional case R. Sharpley [12] obtained sucha decomposition by means of the non-tangential maximal function Ng. Thus, when n = 1we can take Ng in place of Mg on the right-hand side of (1.8). We expect the same resultto be true for n>1 as well.The article is organized as follows. In Section 2 we consider applications of our mainresult based on Theorem A and inequalities (1.4), (1.5); some of them seem to be new. InSection 3 we prove two auxiliary lemmas, and in Section 4 we prove Theorem 1.2. ApplicationsFirst of all we mention an immediate consequence of Theorem A and Theorem 1.Theorem 2.For any measurable function f with f ∗ (+∞) = 0 and all weights ω,( ) p∫R |f |p ωdx ≤ c p,n M∫R # n n λ nf M [p]+1 ωdx (1

468 A.K. LernerCombining this result with (1.5) yeilds the following versions of the Fefferman–Steintheorem (cf. [5]) for arbitrary weights.Theorem 3.For any locally integrable function f with (Mf ) ∗ (+∞) = 0 and all weights ω,(∫R |Mf |p ωdx ≤ c p,n f∫R #) pM [p]+1 ωdx (1

Weighted Norm Inequalities for the Local Sharp Maximal Function 469Corollary 1.Inequalities (2.1) and (2.3) are sharp in the sense that M [p]+1 ω cannot be replacedby M [p] ω.To prove Proposition 1, we follow a counterexample due to J.M. Wilson [16]. Letn = 1 and T be the Hilbert transform H . Take ω = χ (0,1) and f = (log x) −1 χ (e,e N ) .Simple computations show that |Hf |≥c log N on (0, 1),M k ω(x) ≍(log(2 +|x|))k−12 +|x|,andTherefore,while(M r f(x)≤ c1log(2 +|x|) χ {|x|≤e N } + eN/rN∫R |Hf |p ωdx ≥ c(log N) p ,)1|x| 1/r χ {|x|≥e N } .∫∫ e NR (M rf ) p M [p] dxωdx ≤ c0 (log(2 + x)) p−[p]+1 (2 + x)∫+ c eNp/r ∞(log x) [p]−1N p e N x 1+p/r dx≤ c(log N + 1/N p−[p]+1 ) ≤ c log N,which contradicts (2.7) for large N.In conclusion we observe that (2.5) allows us to obtain a sufficient condition on a pairof weights (ω, v) for which T is bounded from L p v to L p ω, that is,∫∫R |Tf|p ωdx ≤n R |f |p vdx. (2.8)nIndeed, applying the well-known Sawyer’s S p condition [11] to (2.5), we get that thecondition∫ ( ( )) ∫pM v 1−p′ χ Q (Mω/ω) p ωdx ≤ c v 1−p′ dx for all Q ⊂ R nQQis sufficient for (2.8) (see also [3, 15] for different results in this direction).3. Two LemmasHere we present two important properties of the local sharp maximal function. Recallfirst some well-known definitions.The non-increasing rearrangement of a measurable function f is defined byf ∗ (t) = sup|E|=t x∈Einf |f(x)| (0

470 A.K. LernerNext, for any measurable f define its median value m f (Q) over Q as a, possiblynonunique, real number such that|{x ∈ Q : f(x)>m f (Q)}|≤|Q|/2 ,|{x ∈ Q : f(x)

472 A.K. Lerner4. Proof of the Main ResultProof of Theorem 1. Since Mλ # |f |≤M# λf , we can assume that f ≥ 0. Supposealso that, additionally, f ∈ L ∞ ,ω ∈ L 1 ∩ L ∞ , and ω ≤ 2 m . Next, we use a standard“atomic decomposition” of ω based on the Calderón–Zygmund decomposition. For k ∈ Zwe write k ={x : M ω(x) > 2 k } as a disjoint union of dyadic cubes Q k jsuch that2 k 2 k }∣∣f − f Q kj∣ ∣∣ dx +∫R n fh −l dx∫M λ # nfdx+R fh n −l dx∫M λ # nfdx+R fh n −l dx .Since ‖h k ‖ 1 =‖ω‖ 1 for all k, the last integral can be estimated for any ε>0by∫∫R fh n −l dx ={x:f>ε}∫fh −l dx + fh −l dx{x:f ≤ε}≤ 2 n 2 −l ‖f ‖ ∞ |{x : f>ε}| + ε‖ω‖ 1 ,

Weighted Norm Inequalities for the Local Sharp Maximal Function 473which clearly gives that ∫ R n fh −l → 0asl →∞. Thus, letting l →∞, we obtain∫R n fωdx ≤ 24 · 2n m−1∑≤ 48 · 2 n2 k ∫k=−∞{M ω>2 k }m−1∑∫ 2 kk=−∞∫ ∞ ∫≤ 48 · 2 n02 k−1 ∫{Mω>α}{Mω>α}M # λ nfdxM # λ nfdxdαM # λ nfdxdα= 48 · 2 n ∫R n M# λ nf(x)Mω(x)dx . (4.1)Next we note that additional assumptions on f and ω can be easily removed. Indeed,in the general case set f j = min(f, j) and ω j = (min(ω, j))χ B(0,j) , where B(0,j)is theball of radius j centered at the origin. Then, (4.1) and Lemma 1 yield∫R n f j ω j dx ≤ 48 · 2 n ∫R n M# λ n(f j )(x)M(ω j )(x) dx≤ 96 · 2 n ∫R n M# λ nf(x)Mω(x)dx .Now the Fatou convergence theorem completes the proof.AcknowledgmentsThe author is grateful to E. Liflyand for useful discussions about the subject of thisarticle.References[1] Chang, S.-Y.A., Wilson, J.M., and Wolff, T. (1985). Some weighted norm inequalities concerning theSchrödinger operator, Comm. Math. Helv., 60, 217–246.[2] Chanillo, S. and Wheeden, R.L. (1987). Some weighted norm inequalities for the area integral, IndianaUniv. Math. J., 36, 277–294.[3] Cruz-Uribe, D. and Pérez, C. (1999). Sharp two-weight, weak-type norm inequalities for singular integraloperators, Math. Res. Lett., 6, 417–427.[4] Cruz-Uribe, D. and Pérez, C. (2000). Two-weight extrapolation via the maximal operator, J. Funct. Anal.,174, 1–17.[5] Fefferman, C. and Stein, E.M. (1972). H p spaces of several variables, Acta Math., 129, 137–193.[6] Jawerth, B. and Torchinsky, A. (1985). Local sharp maximal functions, J. Approx. Theory, 43, 231–270.[7] Journé, J.-L. (1983). Calderón–Zygmund operators, pseudo-differential operators and the Cauchy integralof Calderón, Lecture Notes in Math., 994, Springer-Verlag, Berlin.[8] Lerner, A.K. (1998). On weighted estimates of non-increasing rearrangements, East J. Approx., 4, 277–290.[9] Pérez, C. (1994). Weighted norm inequalities for singular integral operators, J. London Math. Soc., 49,296–308.[10] Pérez, C. (2000). Sharp weighted inequalities for the vector-valued maximal function, Trans. Am. Math.Soc., 352, 3265–3288.

474 A.K. Lerner[11] Sawyer, E.T. (1982). A characterization of a two-weight norm inequality for maximal operators, StudiaMath., 75, 1–11.[12] Sharpley, R. (1986). On the atomic decomposition of H 1 and interpolation, Proc. Am. Math. Soc., 97,186–188.[13] Stein, E.M. (1993). Harmonic Analysis, Princeton University Press, Princeton.[14] Strömberg, J.-O. (1979). Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, IndianaUniv. Math. J., 28, 511–544.[15] Treil, S., Volberg, A., and Zheng, D. (1997). Hilbert transform, Toeplitz operators and Hankel operators,and invariant A ∞ weights, Rev. Mat. Iberoam., 13, 319–360.[16] Wilson, J.M. (1989) Weighted norm inequalities for the continuous square functions, Trans. Am. Math.Soc., 314, 661–692.Received December 13, 2002Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israele-mail: aklerner@netvision.net.il